# Further Properties of Tsallis Entropy and Its Application

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## Abstract

**:**

## 1. Introduction

## 2. Properties of Tsallis Entropy

**Definition 1.**

**Theorem 1.**

**Proof.**

**Theorem 2.**

**Theorem 3.**

**(i)**- ${X}_{1}$ is DFR, then ${H}_{\alpha}\left({X}_{1}\right)\le {H}_{\alpha}\left({X}_{2}\right)$ for all $0\le \alpha \le 1;$
**(ii)**- ${X}_{2}$ is DFR, then ${H}_{\alpha}\left({X}_{1}\right)\le {H}_{\alpha}\left({X}_{2}\right)$ for all $\alpha \ge 1.$

**Proof.**

## 3. Properties of Residual Tsallis Entropy

**Definition 2.**

**Lemma 1.**

**Remark 1.**

**Theorem 4.**

**Proof.**

**Example 1.**

**Theorem 5.**

**Proof.**

## 4. Tsallis Entropy of Coherent Structures and Mixed Structures

**Proposition 1.**

**Proof.**

**Example 2.**

**Theorem 6.**

**Proof.**

**Example 3.**

**Theorem 7.**

**Proof.**

#### Bounds for the Tsallis Entropy of Mixed Systems

**Theorem 8.**

**Proof.**

**Theorem 9.**

**Proof.**

**Example 4.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] [Green Version] - Tsallis, C. Possible generalization of boltzmann-gibbs statistics. J. Stat. Phys.
**1988**, 52, 479–487. [Google Scholar] [CrossRef] - Kumar, V.; Taneja, H. A generalized entropy-based residual lifetime distributions. Int. J. Biomath.
**2011**, 4, 171–184. [Google Scholar] [CrossRef] - Nanda, A.K.; Paul, P. Some results on generalized residual entropy. Inf. Sci.
**2006**, 176, 27–47. [Google Scholar] [CrossRef] - Wilk, G.; Wlodarczyk, Z. Example of a possible interpretation of tsallis entropy. Phys. A Stat. Mech. Its Appl.
**2008**, 387, 4809–4813. [Google Scholar] [CrossRef] [Green Version] - Zhang, Z. Uniform estimates on the tsallis entropies. Lett. Math. Phys.
**2007**, 80, 171–181. [Google Scholar] [CrossRef] - Toomaj, A.; Doostparast, M. A note on signature-based expressions for the entropy of mixed r-out-of-n systems. Nav. Res. Logist.
**2014**, 61, 202–206. [Google Scholar] [CrossRef] - Toomaj, A. Renyi entropy properties of mixed systems. Commun. Stat.-Theory Methods
**2017**, 46, 906–916. [Google Scholar] [CrossRef] - Toomaj, A.; Crescenzo, A.D.; Doostparast, M. Some results on information properties of coherent systems. Appl. Stoch. Model. Bus. Ind.
**2018**, 34, 128–143. [Google Scholar] [CrossRef] - Toomaj, A.; Zarei, R. Some new results on information properties of mixture distributions. Filomat
**2017**, 31, 4225–4230. [Google Scholar] [CrossRef] - Baratpour, S.; Khammar, A. Tsallis entropy properties of order statistics and some stochastic comparisons. J. Stat. Res. Iran
**2016**, 13, 25–41. [Google Scholar] [CrossRef] [Green Version] - Shaked, M.; Shanthikumar, J.G. Stochastic Orders; Springer Science and Business Media: Berlin/Heidelberg, Germany, 2007. [Google Scholar]
- Arnold, B.C.; Balakrishnan, N.; Nagaraja, H.N. A First Course in Order Statistics; SIAM: New York, NY, USA, 2008. [Google Scholar]
- Ebrahimi, N.; Soofi, E.S.; Soyer, R. Information measures in perspective. Int. Stat. Rev.
**2010**, 78, 383–412. [Google Scholar] [CrossRef] - Ebrahimi, N.; Maasoumi, E.; Soofi, E.S. Ordering univariate distributions by entropy and variance. J. Econom.
**1999**, 90, 317–336. [Google Scholar] [CrossRef] - Samaniego, F.J. System Signatures and Their Applications in Engineering Reliability; Springer Science and Business Media: Berlin/Heidelberg, Germany, 2007; Volume 110. [Google Scholar]
- Navarro, J.; del Aguila, Y.; Asadi, M. Some new results on the cumulative residual entropy. J. Stat. Plan. Inference
**2010**, 140, 310–322. [Google Scholar] [CrossRef] - Bagai, I.; Kochar, S.C. On tail-ordering and comparison of failure rates. Commun. Stat.-Theory Methods
**1986**, 15, 1377–1388. [Google Scholar] [CrossRef]

**Figure 1.**The residual Tsallis entropy of order $\alpha =0.5$ (left panel) and $\alpha =2$ (right panel) with respect to t given in Example 1.

Distribution | $\mathit{f}\left(\mathit{x}\right)$ | ${\mathit{H}}_{\mathit{\alpha}}\left(\mathit{f}\right)$ |
---|---|---|

Uniform | $\frac{1}{\beta},\phantom{\rule{4pt}{0ex}}0<x<\beta $ | $\frac{{b}^{-\alpha}-1}{1-\alpha},\phantom{\rule{4pt}{0ex}}b>0$ |

Gamma | $\frac{\lambda}{\mathsf{\Gamma}\left(k\right)}{x}^{k-1}{e}^{-\lambda x},\phantom{\rule{4pt}{0ex}}x>0$ | $\frac{1}{1-\alpha}\left[\frac{{\lambda}^{\alpha -1}\mathsf{\Gamma}(\alpha (k-1)+1)}{{\left(\alpha \lambda \right)}^{\alpha (k-1)}{\mathsf{\Gamma}}^{\alpha}\left(k\right)}-1\right],\phantom{\rule{4pt}{0ex}}k>0$ |

Weibull | $\frac{k}{\lambda}{\left(\frac{x}{\lambda}\right)}^{k-1}{e}^{-{(x/\lambda )}^{k}},\phantom{\rule{4pt}{0ex}}x>0$ | $\frac{1}{1-\alpha}\left[{\left(\frac{k}{\lambda}\right)}^{\alpha -1}\frac{\mathsf{\Gamma}\left(\alpha \right(k-1)-k+2)}{{\alpha}^{\alpha (k-1)-k+2}}-1\right],\phantom{\rule{4pt}{0ex}}k>0$ |

Beta | $\frac{1}{B(a,b)}{x}^{a-1}{(1-x)}^{b-1},\phantom{\rule{4pt}{0ex}}0<x<1$ | $\frac{1}{1-\alpha}\left[\frac{B\left(\alpha \right(a-1)+1,\alpha (b-1)+1)}{{B}^{\alpha}(a,b)}-1\right],\phantom{\rule{4pt}{0ex}}a,b>0$ |

**Table 2.**Comparisons of Tsallis entropy and the lower bound of ${T}_{\alpha}$ for some values of $\alpha $.

N | p | ${\mathit{H}}_{0.5}\left(\mathit{T}\right)$ | ${\mathit{H}}_{1.5}\left(\mathit{T}\right)$ | ${\mathit{H}}_{3}\left(\mathit{T}\right)$ | ${\mathit{H}}_{0.5}^{\mathit{L}}\left(\mathit{T}\right)$ | ${\mathit{H}}_{1.5}^{\mathit{L}}\left(\mathit{T}\right)$ | ${\mathit{H}}_{3}^{\mathit{L}}\left(\mathit{T}\right)$ |
---|---|---|---|---|---|---|---|

1 | (1) | 2.0000 | 0.6666 | 0.4000 | 2.0000 | 0.6666 | 0.4000 |

2 | (1,0) | 0.8284 | 0.1143 | −0.1666 | 0.8284 | 0.1143 | −0.1666 |

3 | (0,1) | 2.4428 | 0.8892 | 0.4333 | 2.4428 | 0.8892 | 0.4333 |

4 | (1,0,0) | 0.3094 | −0.3094 | −1.0000 | 0.3094 | −0.3094 | −1.0000 |

5 | (1/3,2/3,0) | 1.1111 | 0.3948 | 0.2023 | 1.9047 | 0.6309 | 0.2333 |

6 | (0,1,0) | 1.2659 | 0.5069 | 0.2857 | 1.2659 | 0.5069 | 0.2857 |

7 | (0,2/3,1/3) | 2.0698 | 0.7412 | 0.3845 | 1.7169 | 0.6527 | 0.3392 |

8 | (0,0,1) | 2.6188 | 0.9442 | 0.4464 | 2.6188 | 0.9442 | 0.4464 |

9 | (1,0,0,0) | 0.0000 | −2.1666 | −2.1666 | 0.0000 | −2.1666 | −2.1666 |

10 | (1/2,1/2,0,0) | 0.5030 | −0.0375 | −0.3242 | 0.3603 | −0.2260 | −1.0515 |

11 | (1/4,3/4,0,0) | 0.6407 | 0.1431 | −0.0121 | 0.5405 | −0.0057 | −0.4939 |

12 | (1/4,7/12,1/6,0) | 0.9281 | 0.2788 | 0.1081 | 0.6644 | 0.0608 | −0.4471 |

13 | (1/4,1/4,1/2,0) | 1.2514 | 0.4917 | 0.2732 | 0.9122 | 0.1941 | −0.3536 |

14 | (0,1,0,0) | 0.7206 | 0.2145 | 0.0636 | 0.7206 | 0.2145 | 0.0636 |

15 | (0,5/6,1/6,0) | 0.9768 | 0.3318 | 0.1601 | 0.8445 | 0.2811 | 0.1103 |

16,17 | (0,2/3,1/3,0) | 1.1415 | 0.4292 | 0.2333 | 0.9684 | 0.3478 | 0.1571 |

18,19 | (0,1/2,1/2,0) | 1.2659 | 0.5069 | 0.2857 | 1.0924 | 0.4144 | 0.2038 |

20,21 | (0,1/3,2/3,0) | 1.3612 | 0.5646 | 0.3199 | 1.2163 | 0.4810 | 0.2506 |

22 | (0,1/6,5/6,0) | 1.4299 | 0.6012 | 0.3385 | 1.3402 | 0.5477 | 0.2974 |

23 | (0,0,1,0) | 1.4641 | 0.6143 | 0.3441 | 1.4641 | 0.6143 | 0.3441 |

24 | (0,1/2,1/4,1/4) | 2.0179 | 0.6950 | 0.3593 | 1.4044 | 0.5031 | 0.2307 |

25 | (0,1/6,7/12,1/4) | 2.1010 | 0.7655 | 0.3958 | 1.6522 | 0.6364 | 0.3242 |

26 | (0,0,3/4,1/4) | 2.0979 | 0.7653 | 0.3959 | 1.7761 | 0.7030 | 0.3709 |

27 | (0,0,1/2,1/2) | 2.4161 | 0.8751 | 0.4290 | 2.0882 | 0.7917 | 0.3978 |

28 | (0,0,0,1) | 2.7123 | 0.9691 | 0.4515 | 2.7123 | 0.9691 | 0.4515 |

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Alomani, G.; Kayid, M.
Further Properties of Tsallis Entropy and Its Application. *Entropy* **2023**, *25*, 199.
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Alomani G, Kayid M.
Further Properties of Tsallis Entropy and Its Application. *Entropy*. 2023; 25(2):199.
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Alomani, Ghadah, and Mohamed Kayid.
2023. "Further Properties of Tsallis Entropy and Its Application" *Entropy* 25, no. 2: 199.
https://doi.org/10.3390/e25020199